Existence of Steady State Solutions with Finite Energy for the Magnetohydrodynamic Equations in the Whole Space R3

Abstract

We study the steady state Magnetohydrodynamic (MHD) equations in the whole space Following the work of C. Bjorland and M. Schonbek (4) onNavier -Stokes equations in the whole space, we prove the existence of atleast one solution with finite Dirichlet Integral to steady state Magnetohydrodynamic equations in the whole space. Further, we show that these solutions are unique among all solutions with finite energy and finite DirichletIntegral . I. Introduction Magnetohydrodynamics (MHD) is the study of flows of fluids which are electrically conducting and move in a magnetic field. The simplest example of an electrically conducting fluid is a liquid metal like mercury or liquid sodium. MHD treats, in particular, conducting fluids either in liquid form or gaseous form.The equations describing the motion of a viscous incompressible conducting fluid moving in a magnetic field are derived by coupling Navier-Stokes equations with Maxwell's equations together with expression for the Lorentz force. The domain Ω in which the fluid is moving is either a bounded subset of or the whole space . In this paper we restrict our considerations to a domain Ω which is the whole space .During past four or five decades, there have been an extensive study of qualitativeproperties such as existence, uniqueness, regularity and stability of solutions of theMHD equations. This is evident from the work of Duvaut and Lions (1), E. SanchezPalencia (2), Sermange and Temam (3) and other researchers working in the field. Themethods from nonlinear functional analysis such as Galerkin approximation, fixed pointtheorems, monotone and coercive operators, semigroup theory etc have been applied toestablish many a qualitative properties for compressible as well as incompressible MHDflows. The function spaces used are either Holder spaces or Sobolev spaces which are theappropriate function spaces for using these methods and the theory of elliptic operators. In spite of these works, there are very few qualitative results available in the case where the domain is the full space. In the case when domain is a bounded subset of R 3 , it is easy to obtain qualitative results by using Poincare type inequality. But for unbounded domain, one has to use other techniques as were developed by C. Bjorland and M. Schonbek (4). As for MHD flows for incompressible conducting fluids, there are other works where regularity results for MHD flows have been proved ( see references (5-7) ). However, as in the case of Navier-Stokes equations for incompressible fluids, the proof of global regularity remains illusive in this case also. In the present paper, we show that the techniques used in (4) can be extended to prove similar results for steady state Magnetohydrodynamic (MHD) flows. Thus, we consider viscous incompressible Magnetohydrodynamic (MHD) flow governed by the following equations: